The structures and decompositions of symmetries involving idempotents

Abstract: Let $\mathcal{H}$ be a separable Hilbert space and $P$ be an idempotent on $\mathcal{H}.$ We denote by $$\Gamma_{P}={J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and }\hbox{ } JPJ=I-P}$$ and $$\Delta_{P}={J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and }\hbox{ } JPJ=I-P^*}.$$ In this paper, we first get that symmetries $(2P-I)|2P-I|^{-1}$ and $(P+P^{}-I)|P+P^{}-I|^{-1}$ are the same. Then we show that $\Gamma_{P}\neq\emptyset$ if and only if $\Delta_{P}\neq\emptyset.$ Also, the specific structures of all symmetries $J\in\Gamma_{P}$ and $J\in\Delta_{P} $ are established, respectively. Moreover, we prove that $J\in\Delta_{P}$ if and only if $\sqrt{-1}J(2P-I)|2P-I|^{-1}\in\Gamma_{P}.$